LEL-a Newly Designed Molecular Descriptor

Transcription

1 410 Acta Chim. Slov. 2009, 56, Scientific paper LEL-a Newly Designed Molecular Descriptor Dragan Stevanovi}, a,b Aleksandar Ili}, a Cristina Onişor c and Mircea V. Diudea c, * a Faculty of Sciences and Mathematics, University of Ni{, Vi{egradska 33, Ni{, Serbia b University of Primorska - FAMNIT, Glagolja{ka 8, 6000 Koper, Slovenia c Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, Arany Janos Str. 11, , Cluj, Romania * Corresponding author: diudea@gmail.com Received: Abstract A correlating study of topological indices TIs provided by TOPOCLUJ software package and LEL, a newly proposed index built up on the eigenvalues of Laplacian matrix, on thirteen properties of octanes, revealed good correlating ability of a dozen selected TIs, all related to the Wiener index, and of LEL as well. LEL describes well the properties which are well accounted by the majority of the selected molecular descriptors: octane number MON, entropy S, volume MV, or refraction MR, particularly the acentric factor AF parameter, but also more difficult properties like boiling point, melting point and logp. LEL is the best correlated with the WK (Wiener-type number, taken the reciprocal of entries in the combinatorial D p matrix, of higher rank, calculated by TOPOCLUJ software) indices. In a second set of polycyclic aromatic hydrocarbons, LEL was proved to be as good as the Randi} χ index (a connectivity index) and better than the Wiener index (a distance based index). In addition, it is well defined mathematically and shows interesting relations in particular classes of graphs, these recommending LEL as a new, powerful topological index. The actual study proved the considered TIs are basic topological descriptors in prediction of various molecular properties, with good perspective in QSPR/QSAR studies. Keywords: Topological indices, QSPR, QSAR, octanes, polycyclic aromatic hydrocarbons 1. Introduction Elucidation of the relationship between molecular structures and their properties is a challenge to chemists for more than a century. A chemical structure can be quantified in various ways, one of the most popular, in the last decade being that which makes use of Graph Theory, particularly topological indices TIs, which are numerical descriptors encoding topological attributes of a molecular graph. They were used both in graph discriminating analysis as well as in quantitative structure-property relationship QSPR studies for modeling a variety of physico-chemical properties or in quantitative structureactivity relationship QSAR studies to predict biological activities. Nowadays, their number becomes uncountable, as a consequence of the explosive development of computational technology. Randi} 1 has outlined some desirable attributes for the topological indices in the view of preventing their hazardous proliferation, among which: direct structural interpretation, good correlation with at least one property, good discrimination of isomers, simplicity, locally defined, and generalized to higher analogues, are the most important. There are several commercial software packages which calculate more than one thousand TIs: POLLY, 2 MOLCONN, 3 CODESSA, 4,5 DRAGON 6 and TOPOCLUJ. 7 Alkanes represent an interesting class of compounds as a starting point for the application of molecular modeling procedures. Many properties of the alkanes vary function of molecular mass or branching, and alkanes can be described by using a single type of (carbon) atom. There are properties well accounted by a single molecular descriptor, e.g., octane number MON, entropy S, volume MV, refraction MR, etc. Other properties, such as, boiling point BP, heat of vaporization HV, total surface area TSA, partition coefficient LogP, density DENS, critical tempe-

2 Acta Chim. Slov. 2009, 56, rature CT, critical pressure CP and heat of formation DHF are notable exceptions, being not well modeled by any of the parameter sets. Among the well modeled parameters, the acentric factor AF appears to be correlated more than 90% by eleven of twelve descriptors herein discussed. Definition of the best scored TIs, e.g., LEL, 2WD, 2WH, 1WK, 2WUCJD and PDS3[Sh[SZ]] will be given in the following section. The purpose of the present report is to evaluate the relative performances of a pool of descriptors in relating the hydrocarbon molecular structures to a set of physical properties. It is of particular usefulness to predict values of some molecular properties, namely those which are difficult to measure or show health risk or for unavailable substances. In this respect, the newly designed index LEL and those provided by the TOPOCLUJ software are of basic importance. The sets of studied molecules were selected among the representative and sufficiently complex structures (octane isomers and polycyclic aromatic hydrocarbons (PAH)) and the correlations used were in monovariate regression, in view of a more direct interpretation of the results. 2. Description of Indices In any process of molecular modeling, either quantum or correlating one, the need for a representation of molecular structure is critical and its role is significant to find appropriate predictive models. An information rich representation which is rapidly computed and readily manipulated is essential. This is the case of the topological indices, which are among the most used molecular descriptors. TIs are single number descriptors associated with a molecular graph representing a molecule, which does not depend on the numbering and pictorial representation of that graph. In this section, definitions for LEL and the best scored TIs, among the indices provided by the TOPOCLUJ software, are presented LEL an Index Built on the Laplacian Matrix Let G = G(V,E) be a finite, undirected graph with n vertices V = {1,2,...,n} and m = E edges. The degree of a vertex u in V is denoted by d u. Let G have the adjacency matrix A with eigenvalues λ 1 λ 2... λ n, and Laplacian matrix L = D-A, where D is the diagonal matrix of vertex degrees, with eigenvalues μ 1 μ 2... μ n. The Laplacianlike energy, shortly LEL, of G is defined 8 as: (1) It has been shown to have a nice mathematical behavior. It is closely related to the coefficients c k of the characteristic polynomial of the Laplacian matrix L: In particular, 9 for two graphs G and H of order n, if c k (G) c k (H) for k = 1,...n-1, then LEL(G) LEL(H). Furthermore, if a strict inequality c k (G) < c k (H) holds for at least one value k, then LEL(G) < LEL(H). Using this relation, it has been proved 9 that, among trees of order n, the star S n has the minimum, while the path P n has the maximum LEL. Similarly, among unicyclic graphs of order n, the star with an edge between two of its leaves has the minimum LEL, and the cycle C n has the maximum LEL Walk Indices or Wiener-type Indices of Higher Rank A walk w(1,n) in a graph G = G(V,E) (with V = V(G) being the set of vertices and E = E(G) the set of edges) is an alternating sequence of vertices and edges, w(1,n) = (v 1, e 1, v 2, e 2,..., v n 1, e m, v n ), v i V(G ), e i E(G ), m n 1,so that any two subsequent edges are adjacent: (v i 1, v i ) E(G). Revisiting vertices and edges is allowed. If V(w(1,n)) = {v 1, v 2,..., v n 1, v n } is the set of vertices of the walk w(1,n) and E(w(1,n)) = {e 1, e 2,..., e m 1, e m } the set of its edges, then l(w(1,n)) = E(w 1,n ) represents the length of walk w(1,n), which equals the number of traversed edges. If no other condition is imposed, the walk is called a random walk. If the walk starts and ends in the same vertex v n = v 1 it is a closed walk, else it is open Walks of length e, starting from the vertex i V (G) can be counted by summing the entries in the i th row of the e th power of the adjacency matrix A: e W i is called the walk degree (of rank e) of vertex i (or atomic walk count 18,19 ). Local and global invariants based on walks in graph were considered for correlating with physico-chemical properties. 19 Weighted walk degrees can be easily calculated by means of the algorithm proposed by Diudea et al. 20 It evaluates a local (topological) property by iterative summation of vertex contributions over all vertices in row i: where M is a square matrix and e W M is the diagonal matrix of walk degrees (weighted by the property encoded by (2) (3) (4) (5) (6)

3 412 Acta Chim. Slov. 2009, 56, the matrix M). The diagonal entries [ e W M ] i,i represent the row sum of entries in the matrix M raised at power e [M e ] i,j, or the (weighted by M) walk degrees e W M,i : The partition of G with respect to i will be: (10) (7) The half sum of all local invariants e W M,i in G defines a global invariant called the walk number e W M : with ecc i being the eccentricity of i (i.e., the largest distance from i to the other vertices of G). The entries in the layer matrix (of vertex property) LM, is defined as: (11) When M = A, then e W A represents the molecular walk count: 18 when M = D (distance matrix), then e W D is just the Wiener index, 21 of rank e. The extension of this idea resulted in Wiener-type indices of higher rank; 19 the info matrix M can be any square topological matrix. Among such matrices, the following ones are discussed in the present paper: D, H (of reciprocal topological distance entries, also called Harary matrix), 16,17,22 K (of reciprocal D p matrix entries) 16,17,23 and UCJD (the unsymmetrical Cluj matrix on distances) For definition of these matrices and derived indices the reader is invited to consult the refs. 16,17, Indices Designed on Layer/shell Matrices Define a layer of vertices located at distance k to the vertex i as: 19,28,29 (8) (9) with the most used operation being the summation. The zero columns are just the column of vertex properties [LM] i,0 = p i. Any atomic/vertex property can be considered as p i. More over, any square matrix M can be taken as info matrix, in supplying local/vertex properties as row sum RS, column sum CS or diagonal entries given by the Walk matrix. 16,17 Layer matrix is a collection of the above defined entries: (12) with d(g) being the diameter of the graph (i.e., the largest distance in G). Define the entries in the shell matrix (of pair vertex property) SM as: 29 (13) with the most used operation being the summation. Shell matrix is a collection of the above defined entries: (14) Table 1. Topological indices for octanes Molecule LEL 1WD 2WD 1WW 1WH 2WH 1WK 2WK 1/1WK 1/2WK 2WUCJD PDS3[Sh[Sz]]

4 Acta Chim. Slov. 2009, 56, The zero column, [SM] i,0 = 1, in case of zero diagonal square info matrix but any other pair vertex property (written as diagonal entries) can be considered. The index PDS3 [Sh[SZ]] in Table 1, is calculated as columns sum up to distance three on the shell matrix of Szeged matrix, taken as info matrix. For Szeged matrices and indices the reader is invited to consult refs. 16,17,27 The TOPOCLUJ software package 7 is designed to calculate topological descriptors from topological matrices and/or polynomials. Several weighting schemes including group electronegativity, group mass and partial charges are enabled. Topological indices derived from the matrices: adjacency, connectivity, distance, detour, distancepath, detour-path, Cluj, their reciprocal matrices, walkmatrices, walk-operated matrices, layer- and shell-matrices were successfully used in correlating studies and graph discriminating analysis during the last decade. 17,30 The values of the best scored TIs for octane isomers are listed in Table 1. Data for the octanes properties and correlation coefficients, in monovariate linear regression, with the best scored topological indices, among those provided by TOPOCLUJ software, are shown in Table 2. The inter-correlation of indices is presented in Table3 while in Table 4 that of the properties of the octane isomers. Table 2. Data for octanes properties: boiling point BP, motor octane number MON, heat of vaporizat-ion HV, molar volume MV, entropy S, total surface area TSA, acentric factor AF, molar refraction MR, partition coefficient n-octanol/water Log P, density D, critical temperature CT, critical pressure CP and heat of formation DHF and their correlation coefficients, in monovariate linear regression, with the selected topological indices Mol. BP MON HV MV S TSA AF MR LogP DENS CT CP DHF LEL WD WD WW WH WH WK WK /1WK /2WK WUCJD PDS3 [Sh[SZ]] Legend of indices : LEL = Laplacian-like energy 1WD and 2WD = Wiener numbers of higher rank (1 and 2) 1WW = Hyper-Wiener number 1WH and 2WH = Harary numbers of higher rank (1 and 2) 1WK and 2WK = Wiener-type number, taken the reciprocal of entries in the D p matrix, of higher rank (1 and 2). 1/1WK and 1/2WK = global reciprocal of 1WK and 2WK 2WUCJD = Wiener-type number (of rank 2), calculated on Cluj matrix on distance PDS3 [Sh[SZ]] = Columns sum up to distance 3 on the Shell matrix calculated on SZ matrix

5 414 Acta Chim. Slov. 2009, 56, Table 3. Intercorrelation matrix for the best scored indices in octanes Variable LEL 1WD 2WD 1WW 1WH 2WH 1WK 2WK 1/1WK 1/2WK 2WUCJD PDS3 [Sh[SZ]] LEL WD WD WW WH WH WK WK /1WK /2WK WUCJD PDS3[Sh[S]] 1.00 Table 4. Intercorrelation matrix for the selected molecular properties of octanes Variable BP MON HV MV S TSA AF MR LogP DENS CT CP DHF BP MON HV MV S TSA AF MR LogP DENS CT CP DHF 1.00 Data for a second set of 82 polycyclic aromatic hydrocarbons are included in Table 5 while the correlations are given in Table 6. Table 5. Data for polycyclic aromatic hydrocarbons (PAH): melting point MP, boiling point BP, partition coefficient n-octanol/water Log P and the corresponding LEL, Wiener W and Randi} χ indices No. Molecule MP BP LogP LEL W χ 1 naphtalene methylnaphthalene methylnaphthalene ethylnaphthalene ethylnaphthalene dimethylnaphthalene dimethylnaphthalene dimethylnaphthalene dimethylnaphthalene dimethylnaphthalene trimethylnaphthalene trimethylnaphthalene trimethylnaphthalene phenalene phenylnaphthalene phenylnaphthalene anthracene

6 Acta Chim. Slov. 2009, 56, No. Molecule MP BP LogP LEL W χ 18 1-methylanthracene methylanthracene dimethylanthracene dimethylanthracene dimethylanthracene dimethylanthracene phenanthrene methylphenanthrene methylphenanthrene methylphenanthrene methylphenanthrene methylphenanthrene dimethylphenanthrene methylenephenanthrene tetracene benzo[a]anthracene chrysene benzo[c]phenanthrene triphenylene pyrene methylpyrene methylpyrene methylpyrene dimethylpyrene pentacene dibenzo[ai]anthracene dibenzo[ah]anthracene dibenzo[aj]anthracene benzo[b]chrysene dibenzo[ac]anthracene pycene benzo[a]pyrene benzo[e]pyrene perylene coronene anthranthrene benzo[ghi]perylene dibenzo[ae]pyrene methylchrysene methylchrysene methylcholanthrene indeno[1-2-3-cd]pyrene pentaphene hexaphene indano indene azulene acenaphthene acenaphthylene fluorene methylfluorene methylfluorene methylfluorene methylfluorene methylfluorene benzofluorene fluoranthene benzofluorene benzofluorene benzo[ghi]fluoranthene

7 416 Acta Chim. Slov. 2009, 56, No. Molecule MP BP LogP LEL W χ 78 benzo[k]fluoranthene benzo[b]fluoranthene benzo[j]fluoranthene ovalene quaterryllene Table 6. Correlation of PAH properties with selected topological indices Property LEL W χ MP (n = 80) BP (n = 53) LogP (n = 37) Results and Discussion Wiener index is perhaps the most studied topological descriptor. It was used to predict the thermodynamic properties of hydrocarbons since the pioneering works of Wiener. The pool of Wiener-type descriptors is still in growth. Other indices gained considerable attention, and revisited from time to time, to add new and important results. Among these, the Zagreb indices, 31 the Hosoya indices and polynomials 32 and the Cluj indices and polynomials 33 are the best known. In the present work we selected twelve TIs, all but LEL being extensions of the Wiener index. Thirteen properties 34,35 were considered for correlating study, their values in octane isomers and the correlation with the best scored TIs (among more than eighth hundred TIs provided by TOPOCLUJ software package) being listed in Table 1. Other results will refer to a set of aromatic hydrocarbons (see below). Table 2 contains properties from the benchmark data sets of Milano Chemometrics & QSAR Research Group 34, except HV (Heat of Vaporization) which is taken from the Korea Thermophysical Properties Data Bank 35. Nonanes and decanes data are taken from the Korea Thermophysical Properties Data Bank, while polyaromatic hydrocarbons (PAH) are taken from the Milano Chemometrics & QSAR Research Group. All correlations were made in monovariate linear regression, to have a more direct interpretation of the results and a good insight on the structure-property relationship. Boiling point BP, one of the difficult to model properties, is the best correlated by the newly proposed descriptor LEL: R = In n-alkanes, up to eicosane, the Pearson correlation index is R = The index clearly bears size information because in the set of nonanes R = while in decane isomers R = The second well correlated among the selected indices, in the set of octanes, is 1/1WK, (0.599). Compare with the classical Wiener index for which R = Octane number MON, related to the motor combustion, is the best modeled by 2WD, a Wiener-type index of rank 2 ( 0.966), LEL being this time second from the bottom but still highly correlated ( 0.941). Heat of vaporization HV, is the best modeled by LEL (0.887), next being 1/1WK, (0.843). The three above properties are the worst described by PDS3 [Sh[SZ]] (0.432, 0.561, 0.010, respectively). This index, calculated on a shell-type matrix is practically uncorrelated with all the other indices herein discussed and fit the best in case of recalcitrant properties, when the other indices failed (see Table 2, last row, the italicized entries). The matrix of index inter-correlation is given in Table 3. It can be seen that all but the last are related higher than LEL is well correlated with the WK indices. Molecular volume MV, molecular refraction MR and density DENS are highly correlated properties (see Table 4). These are modestly described by the all indices except PDS3 [Sh[SZ]], as above mentioned. Acentric function AF, is the property on which we focus the attention in the following. This parameter is a well described one by all indices excepting the last index. The best score was recorded by 1WK ( 0.995); the newly introduced index LEL is also highly correlated to this parameter in heptanes (0.987), octanes (0.990), nonanes 35 (0.978) and decanes 35 (0.975). A second set of aromatic hydrocarbons 34 (Table 5) was also investigated: melting point MP, boiling point BP and the partition coefficient n-octanol/water LogP, vs. three TIs, LEL, Wiener W and Randi} χ index. 36 The results show LEL as good as χ index and better than Wiener index (Table 6). Within this set of molecules, the degeneracy is comparable for the three indices; LEL is correlated here with the Wiener index W by while with the Randi} index χ by Conclusions A correlating study of topological indices provided by TOPOCLUJ software package and LEL, a newly proposed index, on thirteen properties of octanes revealed good correlating ability of a dozen selected TIs, all related to the Wiener index. LEL is the best correlated with the WK indices. It describes well the properties which are well accounted by the majority of the selected molecular descriptors: octane number MON, entropy S, volume MV, or refraction MR,

8 Acta Chim. Slov. 2009, 56, particularly the AF parameter, but also more difficult properties like boiling point, melting point and logp. Among the desirable attributes required by a good TI, LEL fulfils: good correlation with at least one property, good discrimination of isomers and simplicity. In addition, it is well defined mathematically and shows interesting relations in particular classes of graphs. This index and those provided by the TOPOCLUJ software as well, was proved to be of basic importance in QSAR/QSPR studies. 5. Acknowledgement This work is supported in part by the research program P of the Slovenian Agency for Research and the grants G and of the Serbian Ministry of Science and Technological Development and in part by the Romanian GRANT ID_ References 1. M. Randi}, J. Math. Chem. 1991, 7, S. C. Basak, D. K. Harris, V. R. Magnuson, POLLY 2.3, University of Minnesota, L. H. Hall, MOLCONN and MOLCONN2, Hall Associates Consulting, 2 Davis Street, Quincy, MA A. R. Katritzky, V. S. Lobanov, M. Karelson, CODESSA Manual, University of Florida, Gainesville, Fl, M. Karelson, V. S. Lobanov, A. R. Katritzky, Chem. Rev. 1996, 96, R. Todeschini et al., Dragon software, 7. O. Ursu, M. V. Diudea, TOPOCLUJ 4.0, Babes-Bolyai University, J. Liu, B. A. Liu, MATCH Commun. Math. Comput. Chem. 2008, 59, D. Stevanovi}, MATCH Commun. Math. Comput. Chem. 2009, 61, D. Stevanovi}, A. Ili}, On the Laplacian coefficients of unicyclic graphs, Lin. Algebra Appl. 2009, 430, F. Harary, Graph Theory, Addison-Wesley, Reading, Ma, N. Trinajsti}, Chemical Graph Theory, CRC Press, Inc., Boca Raton, Florida, E. Cayley, Philos. Mag. 1874, 67, C. Berge, Graph Theory and Its Applications (in Romanian), Ed. Tehnică, Bucharest, M. Randi}, W. L. Woodworth, A. Graovac, Int. J. Quant. Chem. 1983, 24, M. V. Diudea, I. Gutman, L. Jäntschi, Molecular Topology, NOVA, New York, M. V. Diudea, M. S. Florescu, P. V. Khadikar, Molecular Topology and Its Applications, EFICON, Bucharest, G. Rucker, C. Rucker, J. Chem. Inf. Comput. Sci. 1993, 33, M. V. Diudea, J. Chem. Inf. Comput. Sci. 1996, 36, M. V. Diudea, M. Topan, A. Graovac, J. Chem. Inf. Comput. Sci. 1994, 34, H. Wiener, J. Amer. Chem. Soc. 1947, 69, M. V. Diudea, J. Chem. Inf. Comput. Sci. 1997, 37, M. V. Diudea, I. Gutman, Croat. Chem. Acta, 1998, 71, M. V. Diudea, MATCH Commun. Math. Comput. Chem., 1997, 35, M. V. Diudea, J. Chem. Inf. Comput. Sci. 1997, 37, M. V. Diudea, B. Prv, M. I. Topan, J. Serb. Chem. Soc. 1997, 62, D. Jane`i~, S. Nikoli}, N. Trinajsti}, Graph Theoretical Matrices in Chemistry, MCM, Kragujevac, M. V. Diudea, J. Chem. Inf. Comput. Sci. 1994, 34, M. V. Diudea, O. Ursu, Indian J. Chem., 42A, 2003, G. Katona, M. V. Diudea, Acta Univ. Cibiniensis, 2002, 5, D. Jane`i~, A. Mili~evi}, S. Nikoli}, N. Trinajsti}, and D. Vuki~evi}, Croat. Chem. Acta 2007, 80, D. Jane`i~, B. Lui~i}, A. Mili~evi}, S. Nikoli}, N. Trinajsti}, and D. Vuki~evi}, Croat. Chem. Acta 2007, 80, M. V. Diudea, A. E. Vizitiu and D. Jane`i~, J. Chem. Inf. Model. 2007, 47, Milano Chemometrics & QSAR Research Group, Molecular Descriptors Dataset (available online at ) 35. Korea Thermophysical Properties Data Bank, (available online at ). 36. X. Li, Y. Shi, MATCH Commun. Math. Comput. Chem. 2008, 59, Povzetek Deskriptorje dobljene s programskim paketom TOPOCLUJ in indeksom LEL osnovanim na Laplaceovi matriki smo uporabili za izgradnjo korelacijskih modelov, ki slu`ijo za opis trinajstih razli~nih lastnosti oktanov. Deskriptor LEL zelo dobro opi{e ve~ino izbranih lastnosti, kot so: oktansko {tevilo, entropija, volumen molekule, indeks refrakcije in parameter AF. Deskriptor LEL je uporaben tudi za opis bolj kompleksnih lastnosti kot sta temperaturi vreli{~a in tali{~a, ter logp. Opazili smo dobro korelacijo indeksa LEL z indeksi WK, izra~unanimi s programom TOPOCLUJ. Ob korelacijski analizi drugega obravnavanega seta spojin policikli~ni aromatski sistemi se je LEL izkazal tako dobro kot Randi}ev χ indeks in bolje kot Wienerjev indeks. Na{a {tudija je pokazala, da je novi topolo{ki indeks LEL uporaben za matemati~ni opis dolo~enih razredov grafov.